Every Non-regular Subcubic Graph is (1,1,2,2)-Packing Colorable

Abstract: For a non-decreasing sequence of integers $S=(s_1,s_2, \dots, s_k)$, an $S$-packing coloring of $G$ is a partition of $V(G)$ into $k$ subsets $V_1,V_2,\dots,V_k$ such that the distance between any two distinct vertices $x,y \in V_i$ is at least $s_{i}+1$, $1\leq i\leq k$. Gastinau and Togni posed the question of whether every subcubic graph admits a $(1,1,2,2)$-packing coloring, except for the Petersen graph, which admits a $(1, 1, 2,2, 3)$-packing coloring. While several studies have investigated this question for special classes of subcubic graphs, a complete solution remains open. In this paper, we make progress toward answering this question by proving that every non-regular subcubic graph is $(1,1,2,2)$-packing colorable.